How crystals that sense and respond to their environments could evolve

How crystals that sense and respondto their environments could evolve

Rebecca Schulman Æ Erik Winfree

Received: 1 September 2006 / Accepted: 26 March 2007 / Published online: 13 June 2007� Springer Science+Business Media B.V. 2007

Abstract An enduring mystery in biology is how a physical entity simple enough to have

arisen spontaneously could have evolved into the complex life seen on Earth today. Cairns-

Smith has proposed that life might have originated in clays which stored genomes con-

sisting of an arrangement of crystal monomers that was replicated during growth. While a

clay genome is simple enough to have conceivably arisen spontaneously, it is not obvious

how it might have produced more complex forms as a result of evolution. Here, we

examine this possibility in the tile assembly model, a generalized model of crystal growth

that has been used to study the self-assembly of DNA tiles. We describe hypothetical

crystals for which evolution of complex crystal sequences is driven by the scarceness of

resources required for growth. We show how, under certain circumstances, crystal growth

that performs computation can predict which resources are abundant. In such cases,

crystals executing programs that make these predictions most accurately will grow fastest.

Since crystals can perform universal computation, the complexity of computation that can

be used to optimize growth is unbounded. To the extent that lessons derived from the tile

assembly model might be applicable to mineral crystals, our results suggest that resource

scarcity could conceivably have provided the evolutionary pressures necessary to produce

complex clay genomes that sense and respond to changes in their environment.

Keywords Evolution � Complexity � Universality � Crystals � Self-assembly �Tiles � Metabolism

R. Schulman � E. Winfree (&)California Institute of Technology, Pasadena, CA 91125, USAe-mail: [email protected]

R. Schulmane-mail: [email protected]

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Nat Comput (2008) 7:219–237DOI 10.1007/s11047-007-9046-8

1 Introduction

Developments in DNA computing have shown that computation can be embedded in

crystal growth processes (Winfree 1996; Rothemund et al. 2004), with potential applica-

tions to combinatorial search problems (Lagoudakis and LaBean 2000; Mao et al. 2000)

and to fabrication tasks in nanotechnology (Cook et al. 2004; Barish et al. 2005). But does

crystal computation have any relevance to what we observe in nature? We speculate here

about a possible connection to the origin of life.

The background for this argument is a hypothesis, proposed and developed by Graham

Cairns-Smith (Cairns-Smith 1966), that the first primitive ‘‘organisms’’ were clay crystals.

In this theory, information (the first ‘‘genes’’) consisted of patterns stored as variations in

crystal structure that could be propagated during crystal growth. For example, in some

layered silicate clays, there are two distinct layer types that appear in a cross-section as a

sequence that could be considered the crystal’s genotype. Replication would occur by

periodic physically-induced fragmentation of crystals into smaller pieces containing the

same genotype, leading to exponential increase in the number of organisms. Cairns-Smith

considered several types of selective pressures that could have been present and would

have resulted in favoring the growth of crystals with non-trivial genotypes. For example,

the structure of a layer sequence could result in different crystal morphologies and different

susceptibilities to fragmentation. He further envisioned the clays interacting with organic

molecules, somehow leading to novel clay-produced organic molecules and eventually to

the genetic takeover of organic life forms (Cairns-Smith 1982).

One of the strengths of Cairns-Smith’s hypothesis is that it is easy to see how Darwinian

evolution could have gotten started by geological processes. However, while clays are

known to interact with organic molecules (Pitsch et al. 1995; Hanczyc et al. 2003), it is

hard to know whether these interactions could have provided evolutionary pressure toward

increasing complexity. It is therefore interesting to ask whether there are other possible

mechanisms that could have stimulated the original evolution of complex sequence

information. In fact, it is hard to envision how anything so simple that it could have arisen

spontaneously could have evolved into the remarkable complexity of form and function

found in modern biological organisms. The capacity of evolutionary systems to create

increasingly complex forms with increasingly adapted function—so-called open-ended

evolution—remains poorly understood. To our knowledge, there are no examples in arti-

ficial life (Bedau et al. 2000), in vitro chemical evolution (Wright and Joyce 1997; Joyce

2004), or in vivo directed evolution (Yokobayashi et al. 2002) that have convincingly

demonstrated open-ended evolution.

The conceptual and physical simplicity of crystal evolution makes it a promising place

to examine these issues. In this paper, we consider whether there are properties of crystal

growth that can lead to open-ended evolution. Examining the capacity for interesting

evolutionary landscapes requires distinguishing between crystal genotype and phenotype—

what is the genetically-determined function performed by crystals that gives rise to a

selective advantage? In this paper, we investigate the ability of crystals to respond to

selection pressures using computations that occur during growth.

The notion that a crystal can perform a computation is based on the observation that the

tiling problem (the question of whether a set of geometric shapes can tile the plane) is

undecidable: any problem solvable by a Turing machine can be expressed as a question of

whether a particular set of shapes can tile the plane (Wang 1962). This observation led to a

constructive method of computing by arranging tiles into a lattice (Winfree 1996). These

220 R. Schulman, E. Winfree

123

tiles can be viewed as analogues for crystal monomers; attachment at a specific site in a

growing crystal is determined by how well a tile’s shape fits in the growth site. The binding

of a particular tile at a particular site can be viewed as a computational or information

transfer step.

It is reasonable to assume that crystals grow in environments where their monomers are

present in solution at different (possibly time-varying) concentrations. In such an envi-

ronment, which crystal patterns grow the fastest? Are there environments in which crystals

must perform computations in order to grow quickly? We show that crystals can sense the

environment and respond by making use of the most abundant tiles. We call this feature a

‘‘crystal metabolism’’ because the computation they perform controls the extraction of

resources from the environment. In particular, we are interested in whether there are

environments that lead to the evolution of increasingly complex crystal metabolisms.

We examine these issues in principle within a previously described tile assembly model

(Winfree 1998), which is a generalized crystal growth model that has been used to study

algorithmic self-assembly of synthetic DNA tiles (Rothemund and Winfree 2000; Rothe-

mund et al. 2004; Barish et al. 2005). It has been previously argued (Schulman and

Winfree 2005) that DNA tile crystals have the capacity for Darwinian evolution of the sort

imagined by Cairns-Smith. Insights derived from studying this model should be applicable

to crystals composed of a variety of other materials, such as proteins (Fygenson et al. 1995;

Collins et al. 2004), RNA (Chworos et al. 2004), or even macroscopic tiles (Bowden et al.

1997; Rothemund 2000) or clay minerals (Cairns-Smith and Hartman 1986).

The tile assembly model describes crystal monomers as square or rectangular tiles with

each unit edge labeled to indicate how it fits with other monomers. Crystal growth pro-

ceeds by accretion, with single tiles being added to the crystal at sites where they make a

sufficient number of contacts. In this paper we consider a version of this model in which (a)

a tile may be added to a site if labels on at least two edges match those presented by the

crystal at that site, and (b) monomer tiles arrive at potential binding sites with a frequency

proportional to their concentration in solution. Occasional violations of rule (a) are referred

to as ‘‘mutations’’. A system consists of a set of tiles, along with the concentrations of

those tiles in solution. Because of the particular choice of matching rules, each tile set

implicitly determines what arrangements of tiles can grow as crystals. If certain arrange-

ments grow faster than others under given conditions, then we consider these arrangements

more fit. Later we will discuss how growth rates relate to the rate of exponential repro-

duction.

To discuss evolution, it is helpful to consider three aspects of an evolutionary process.

First, a self-replicating entity carries information which directs behavior. The space of

achievable behaviors is therefore inherently dependent on the material from which they are

constructed. Second, there must be an environment in which certain behaviors have

selective advantage; this provides the stimulus for evolution to discover complex solutions.

Third, for evolution to proceed quickly there must be a route via mutations in which ever

more complex behavior is achieved by a series of incremental steps. In crystal evolution,

the first aspect (potential for complexity) derives from a choice of a particular tile set; this

determines the behavior implicit in the crystal growth. The second aspect (stimulus for

complexity) derives from the conditions in which the crystals are grown, e.g., tile con-

centrations and temperature, etc. The third aspect (the route to complexity) is difficult to

predict. We will not address it here; for our purposes, it is enough to know that the fittest

crystals could arise through (possibly extremely unlikely) mutations or spontaneous gen-

eration. However, understanding the route to complexity is an important goal for future

studies of crystal evolution.

How crystals that sense and respond to their environments could evolve 221

123

In addition to the above, open-ended evolution seems to require that the selective

pressure provided by the environment is distinct from the potential for evolution, in the

sense that different environments must lead to distinct functional solutions. For example, in

modern biology, the universal potential of biochemistry—the ability of DNA to code for

proteins and other macromolecules that create seemingly arbitrarily sophisticated and

complex machines—makes it possible for living organisms to adapt to an incredible

variety of environmental niches, opening the way to ever-increasing complexity. Thus,

searching for tile sets capable of open-ended evolution, we first see that a tile set plays the

role of a ‘‘chemistry’’ in the sense that it it defines the rules by which tile-based

‘‘organisms’’ can grow and function, and we further expect that we are looking for a tile

set that displays some sort of universality of behavior.

Following this intuition, we argue that tile-based crystals can exploit computation to

enhance their growth rate, and that this can lead to evolutionary processes resulting in

increasing complexity of crystal structure. We introduce a framework for studying the

evolution of metabolic control in crystals under resource-limited growth conditions. Within

this framework, we design tile sets and environments that give rise to evolutionary land-

scapes in which crystals that perform more effective computations are fitter. We provide

two main examples. First, in order to provide simple examples of metabolic evolution, we

describe a tile set that can encode programmable logical computations. Second, to

emphasize that arbitrarily complex computations can in principle be used by crystals to

sense and respond to their environment, we exhibit a tile set capable of universal com-

putation.

2 Zig-zag ribbon evolution

Previously, it was suggested that DNA tile assemblies could in principle evolve through

cycles of crystal growth and splitting (Schulman and Winfree 2005) (Fig. 1). The zig-zag

tile set described in that work produces ribbon-like crystals that copy information along the

length of the crystal. The tile set includes a group of square tiles, and two rectangular tiles

called double tiles. Logical representations of the tiles that comprise a basic zig-zag ribbon

are shown in Fig. 2a.

When growth occurs according to the tile assembly model, tiles are added to a

ribbon in a zig-zag pattern shown in Fig. 2b. Only tiles that match at least two edges

can attach. Given this constraint, the design of the tiles is such that at any moment there

is just one tile that may be added to each end of the ribbon. The addition of each new

row can be viewed as the copying of the information in the previous row. Using the tile

set shown in Fig. 2a, this copying is trivial—only one sequence type is possible.

However, the requirement that a tile attach by two bonds means that it must match both

its vertical neighbor, either above and below, and its horizontal neighbor, to the left or

right. In the case that several tile types may match the vertical neighbor, as shown in

Fig. 2c, only the tile type that already appears in the row will match the horizontal

neighbor. Only this tile can be added, so that information is inherited from layer to

layer. As an example, the tile set shown in Fig. 2c has two tiles for each position and

therefore can propagate one of four strings. This construction can be generalized to an

additional number of bits by adding tiles to the tile set in Fig. 2c—2n sequences can be

propagated by a tile set containing 4n + 2 tiles. Further, a related tile set containing

only six tiles can copy binary sequences of arbitrary width.


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The growth of a crystal increases the number of copies of the original information

present in the ribbon but does not produce any new growth fronts, so copying occurs at a

constant rate. This information copying can be accelerated by periodic forces that cause

ribbons to break. For each new ribbon that is created by breakage, two new growth fronts

become available. Repeated fragmentation will therefore exponentially amplify an initial

piece of information. Occasionally, a tile matching only one bond rather than two will join

the assembly, resulting in a copying error, which will also be inherited. Such copying

errors, inevitable in any physical implementation of tiles (e.g., Winfree 1998), will lead to

evolution if ribbons with certain sequences grow faster than others.

3 Dynamics of crystal evolution

In this section we formulate a simple dynamical model to determine whether crystal

growth and breakage leads to selective amplification, and to elucidate which properties of

the environment and tile set are important in determining the replication rate of a sequence.

The model tracks the concentration of crystals of each possible sequence. For a sequence s,

two parameters are of interest: Fs, the number of growth fronts that can copy sequence s,

and Rs, the number of columns of tiles, totaled over all crystals, with sequence s. The

Fig. 1 The zig-zag crystal life cycle. Zig-zag crystals grow by copying their sequences of DNA tiles.Reproduction occurs when a crystal is broken by external mechanical force (e.g., shearing, as shown in theupper right of the test tube). The small crystals that are the result of division (center) continue growing, andeventually become large enough (top left) to split again. The materials required for growth (tiles) areconstantly replenished by an inward flow, while an outward flow removes slow-growing crystals from thepopulation. Occasional mutations are propagated during growth and are eventually replicated. Similarly,new assemblies are occasionally generated spontaneously (lower left) from single tiles. Once they reach acertain size, these spontaneously generated assemblies can also grow and reproduce


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number of columns, Rs, increases at a rate proportional to the number of growth fronts, Fs,

times the rate at which a new column can be added to a growth front, ks. As this model is

meant to be used in cases where growth rates depend upon the sequences s, ks depends on s,

reflecting the influence of tile set and environment. New growth fronts are produced when

assemblies split into two pieces. We’ll assume that splitting occurs with equal probability

at each column, at a splitting rate ps per column. (Again, this might be sequence-depen-

dent.) Crystals die at rate f by being flushed out of solution.

Assuming that the growth rate is greater than the splitting rate (ks > ps), the dynamics of

this system can be described by two linear differential equations for each sequence.

d

dt¼ Rs

Fs

� �¼ �f ks

ps �f

� �Rs

Fs

� �ð1Þ

(a)

(b)

(c)

Fig. 2 The zig-zag tile set. (a) The basic width 4 zig-zag tile set consists of six tile types. Tiles cannot berotated. The tiles shown here have unique bonds that determine where they fit in the assembly: each labelhas exactly one match on another tile type. (b) The zig-zag tiles are designed to form the assembly shownhere. Tiles can attach to the crystal where they match two edges of the crystal. Two alternating tiles in eachcolumn enforce the placement of the double tiles on the top and bottom, ensuring that growth continues in azig-zag pattern. While growth on the right end of the molecule is shown here, growth occurs simultaneouslyon both ends of the molecule. At each step, a new tile may be added at the location designated by the smallarrow. (c) The tile set shown in Fig. 2a forms only one kind of assembly. The addition of the four tilesshown here allows four types of assemblies to be formed. Each assembly grows by copying its sequence (thevertical cross-section of tiles)


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Because mutations are not included in this model, pairs of equations describing the growth

and replication of a sequence are decoupled from equations describing the dynamics of other

sequences. It is therefore not difficult to solve them; for a given sequence, the eigenvalues of

the solution are �f �ffiffiffiffiffiffiffiffiksps

p: Assuming the death rates for all sequences are the same,

sequences with faster growth rates and higher splitting rates are therefore amplified more than

sequences that grow and split more slowly. As the death rate increases, only the sequences for

whichffiffiffiffiffiffiffiffiksps

pis large remain in solution. These sequences are selected for.

How does crystal evolution compare to RNA or DNA replication, e.g., of viral or

bacterial genomes (Eigen 1971), in which the growth rate is proportional to the concen-

tration of sequences? As the decaying eigenmode dies away, Rs !ffiffiffiks

ps

qFs: (The ratio

ffiffiffiks

ps

qcan be interpreted as half the average length of growing and splitting crystals. The ratio is

halved because each crystal has Rs rows and two growth fronts.) A new equation that

measures only the dynamics of Fs, assuming Rs �ffiffiffiks

ps

qFs; is

d

dtFs ¼

ffiffiffiffiffiffiffiffiksps

pFs � fFs ð2Þ

With the addition of mutation, where a mutation from sequence s to sequence t happens

at rate mst, this model is equivalent to Manfred Eigen’s model of RNA replication (Eigen

1971) with the replication rate of a sequence replaced by the parameterffiffiffiffiffiffiffiffiksps

p:

d

dtFs ¼

ffiffiffiffiffiffiffiffiksps

p� f

� �Fs þ

Xt

mtsFt � mstFsð Þ ð3Þ

Thus, this model is a generalized version of Eigen’s model, with DNA or RNA repli-

cation being the special case where ks = ps. When ks > ps, the fitness (ffiffiffiffiffiffiffiffiksps

p) of a crystal in

a given environment depends on both the growth rate and the splitting rate. Thus, to show

that a particular sequence s is fit, we must therefore show thatffiffiffiffiffiffiffiffiksps

pis large, and that for

unfit t, sequencesffiffiffiffiffiffiffiffiktpt

pis small.

4 A zig-zag ribbon metabolism

By what basis might some zig-zag crystals grow faster than others? In some models of

crystal growth (Winfree 1998), the rate of attachment of a tile to a crystal is proportional to

the concentration of the tile in solution, and the rate at which a tile is removed is related to

the energy loss due to breaking the bonds between the tile and the rest of the crystal. Thus,

the growth rate of crystals can be made faster by increasing the concentration of their

component tiles in solution. Increasing growth rates increases a crystal’s fitness. Similarly,

increasing breakage rates also increases a crystal’s fitness (unless breakage occurs more

often than growth).

In previous examples of zig-zag ribbon evolution (Schulman and Winfree 2005), tile

sets (or ‘‘chemistry’’) needed to be made more complicated in order to achieve more

complex selection. In contrast, in biology a single chemistry has led to the evolution of

more and more complex organisms. Such evolution has occurred because the fitness of a

biological organism is a function of whether its genome contains a program that efficiently

directs resource acquisition, development or relations to other organisms in its environ-

ment, and thus changes in the environment can drive the evolution of complexity (and visa

versa).


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Is there an analog of this mechanism for crystal evolution? That is, is there a single tile

set that allows zig-zag crystals to achieve selective advantage in many different envi-

ronments by performing functions adapted to their environment? While a zig-zag ribbon

cannot direct any function except its own assembly, the fact that the assembly of tile

crystals can perform universal computation (Winfree 1996) suggests that the answer could

be ‘‘yes’’.

An important element of the survival of a self-replicating system in the physical world

is the ability to handle variation in the availability of raw materials needed for growth.

While modern cells use genetic networks and signal transduction in order to respond

optimally to available raw materials, here we describe how zig-zag crystals could evolve a

simple ‘‘metabolism’’ by using tile assembly to compute which tiles to use for growth.

This mechanism, consisting of a set of tile types and their environment, is too complex to

be a model of real crystal growth processes. It is instead intended as a conceptual dem-

onstration that it is possible for zig-zag crystals to evolve complex phenotypes.

We consider a situation where tile resources may be limited and where the addition of a

tile may provide information about the environment. In the examples presented here,

boundary tiles are present at varying concentrations while the concentrations of non-

boundary tiles do not vary.

Two types of boundary tiles, called measurement tiles (Fig. 3a), initiate new rows from

the right during upward growth. Both measurement tiles have the same input edge, but

different measurement tiles have different output edges. Because measurement tiles share

the same input edges, both available measurement tiles can attach to the right edge of the

crystal. The chance that a particular measurement tile will attach is dependent on its

concentration in solution. As will be described below, which measurement tile attaches

determines the output edge that guides proceeding assembly of the crystal. We will assume

that while the relative concentrations of measurement tiles change, their total concentration

stays the same.

Another set of tiles may also become more or less available as time goes on. These are

the resource tiles (Fig. 3a). Changes in the concentrations of resource tiles may be cor-

related with changes in the concentrations of measurement tiles. While both measurement

tiles have the same input edge, each resource tile has a different input edge. Only the

resource tile that matches the left label on the binding site where a resource tile may be

added can fit.

Figure 3a shows a set of ‘‘computation tiles’’ that perform boolean functions. In Sect. 2,

we described how the requirement that a tile match the perimeter of an assembly by two

edges in order to attach allows tile assembly to copy a sequence. A similar principle allows

a crystal to perform local computations that modify the sequence: the two edges by which

the tile attaches to the assembly serve as inputs, and the two edges of the tile that remain

unattached serve as outputs (Winfree 1996). These outputs then serve as inputs for future

computation steps. On the computation tiles shown here, the label on the bottom of each

tile encodes the gate type and an input value, and the label on the top encodes the same

gate type as well as an output value. The left and right edges of the tile encode input and

output values, depending on the direction of assembly. Thus, with each zig and each zag of

growth, the sequence of gate types is copied verbatim from row to row, while simulta-

neously the sequence of boolean values are being processed according to the logic spec-

ified by the gates (Fig. 3b). That is, the computations performed by tile attachment modify

the sequence of boolean values from layer to layer, but the sequence of gate types is

inherited intact. The concentration of the computation tiles remains constant over time.


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The label on the output edge of an attaching measurement tile, either 0 or 1, will serve

as an input for the attachment of the next tile, a computation tile. Conversely, the label that

will be an input edge for a resource tile, either 0 or 1, will be an output of this series of

computations. Thus, the order in which measurement tiles arrive at the right side serves as

input to the computation tiles, which in turn produce outputs consisting of a series of input

edges for resource tiles on the left side.

What kind of assembly is selected for in this environment? Fit assemblies are those

which have the highest replication rate. If it is assumed that all assemblies split at the same

rate, a fit assembly is one that grows quickly. Because the environment changes over time,

the fittest assembly is the one that has a large average growth rate. The rate at which a new

row is added, ks, is the inverse of the total time that is needed to complete a row. The time

needed to add each measurement and computation tile stays constant, because the rate of

tile addition is dependent on tile concentration in solution, and these tiles are present in

constant supply. (While the concentration of the two kinds of measurement tiles vary, their

total concentration remains constant and both types will bind at a given binding site.) The

time needed to bind a resource tile changes, however, and is dependent on the current

concentration of the resource tile. Two observations can be made. First, a row is added

more quickly if fewer computation tiles are used, and second, a row is added more quickly

when the resource tile that is needed to complete the row is abundant. To achieve the first

requirement, an assembly should be as thin as possible. To achieve the second, an assembly

F = (f1, f2, f3, f4)

tilescomputing

tilesmeasurement

tilesresource

L

1>

L

<0

L

L

1>

<1F:

F:

<f4(X,Y) <Y

X

f3(X,Y)

F:

F:

Y> f2(X,Y)>

f1(X,Y)

X

R

R

1>

<0

R

R

1>

<1

(a)

R

R

1>

L<1

LL

L1>

<1

R

R

1>

<1

1>G:

G:1>1

1

F:

F:<0 <0

1

0

F:

F:<1 <1

0

1

F:

F:1> 1>

0

0

F:1>1>

F: 1

1

<1 <0G:

G:1

1

1>G:

G:1>1

1

F = (X and Y, X or Y, not Y, X)G = (X, not Y, 1, not X and Y)

L

L1>

<0G:

G:<1<00

1

<0

1>

(b)

prediction of the next tile

concentrations of resource tiles

computation path

which measurement tile attachesmay foreshadow changes in the

(c)

Fig. 3 Zig-zag crystals which exhibit metabolic control. (a) A tile set for the evolution of metabolic controlcontains computation, resource and measurement tiles. Computation tiles (middle) contain two kinds oflabels: those on the left and right encode a boolean value, either 0 or 1, (shown as X, Y, or f(X,Y)) and thedirection of assembly, either left or right. The labels on the top and bottom encode a gate type (F is shown)and a boolean input and output. Each gate type consists of four boolean functions: those passed to the top onleftward and rightward growth, and those passed to the left and right on leftward and rightward growthrespectively. A computation tile has the same gate type on the top and bottom, thus ensuring that thesequence of gate types is copied from row to row. We assume that gate types corresponding to allcombinations of four boolean functions are provided: there are 8 · 164 such tiles, which is admittedly quitelarge, although similar behavior should be possible with many fewer tile types. Resource tiles are boundarytiles to the left that have the same output value, a 1, but different input values. A crystal must providebinding sites for common resource tiles in order to grow quickly. Measurement tiles have the same input, butdifferent outputs. These outputs can provide growing tile assemblies with information about theenvironment. ‘‘Smart’’ crystals use the information provided by the measurement tiles to predict whichkind of resource tile is most available. (b) An example computation using the gates shown in (a). For eachgate type, the four computations in parenthesis are the four computations for each gate type, in the ordershown on the computation tiles in (a). (c) A large assembly consisting of the tiles in (a). The shading ofcomputation tiles represents their gate type, which is passed from row to row and determines the genotype ofthe ribbon. Squares with dots represent tiles that output 0’s, and squares with no dots represent tiles thatoutput 1’s. The 0’s and 1’s represent the state of a computation, which is updated (but not necessarilycopied) from row to row


123

should correctly predict the abundant resource tile and assembly should produce a binding

site for it rather than for a less abundant resource tile.

How can a program predict the future concentrations of tiles? The assembly of tiles

transforms an input signal, a series of bits received from the output edges of the mea-

surement tiles, to an output signal, a series of binding sites for resource tiles. This trans-

formation depends on the sequence of gate types that is propagated in the crystal. A fit

assembly produces an output signal that is the same as the binding site of the more

abundant resource tile1. For example, when the input and output signal are time varying but

identical, as in Fig. 4a, an output signal that is the same as the input signal accomplishes

this goal. When they are exactly opposite, as in Fig. 4b, inverting the input signal, as is

shown, accomplishes this goal. Thus, the most fit genomes in these environments are the

empty sequence and the inverting gate respectively.

In some environments, an assembly that successfully predicts the identity of the more

abundant resource tile must perform a less trivial computation. Figures 4c and d show

examples, described below, of assemblies that do such computations. While a thinner

assembly may add the computation tiles in its row more quickly, it would often ask for less

abundant resource tiles. Thus, it would spend time waiting to bind these tiles, and therefore

might grow more slowly overall than an assembly with more computation tiles that used

abundant resource tiles.

A tile program that is well adapted to the environment in Fig. 4c, where the output

signal is a time-delayed copy of the input signal, consists of a series of ‘‘delay’’ gates. A

delay gate passes the bit it receives from the previous row to the left and passes the bit

received from the right to the next row. With a program consisting of n delay units, the

crystal will respectively bind a resource tile with a 1 or 0 binding site 2n rows after

receiving a corresponding measurement tile with a 1 or 0 output site. For a longer or

shorter delay, more or fewer delay gates may be used.

When measurement tiles provide no information about the concentrations of resource

tiles, as in Fig. 4d, a successful assembly is one that ignores the information received from

the measurement tiles and computes a set of outputs in a temporal pattern very similar to

the changes in the abundant resource tile. The assembly shown uses its rightmost gate to

discard the input from the measurement tiles. It uses a binary counter program (Cook et al.

2004) (the center two computation tiles) to count to four over and over again. A counter

consists of a series of exclusive or/and gates. The inputs to the counter come from

rightmost digit of the counter (which is a 1 at each iteration) and the bottom edge of the

counter tiles. At each iteration, the counter has two outputs—an output to the left which

signals to the output tile to the left, and a set of outputs that become inputs to the next

iteration of the counter. The ‘‘counter’’ receives its name because these outputs are, read as

a binary number, one larger than the inputs. When the counter reaches its maximum value,

in this case three, a one is output to the left. The left-most gate uses this periodic trigger to

change its requested resource tile. The assembly shown asks for four 1-type resource tiles,

then asks for four 0-type resource tiles, and repeats this cycle. If the requests are in tandem

1 In growth that proceeds downward, rather than upward, the tiles labelled resource tiles function asmeasurement tiles, and vice versa. Thus, both assemblies that can predict the resource tile types that areavailable from the measurement tiles, and those that can predict the measurement tile types that will beavailable from the current concentration of resource tiles will be selected for. Note, however, that gate typesmay be non-deterministic during downward growth, which could result in crystal growth stalling when a tileis incorporated that creates a binding site that matches no gate tile’s outputs. Therefore, consideration ofdownward growth rates is necessary for a complete evaluation of a crystal’s fitness; but we neglect it here tosimplify the presentation.


123

with the periodic change of resource tile type, the assembly will spend little time waiting

for resource tiles, and will grow quickly.

The assemblies containing these programs will spend little time waiting for resource

tiles; thus, they will grow faster than other assemblies of the same size that must wait for

the right resource tile to become available. But will they grow faster than thinner

assemblies, which don’t have to spend time adding computation tiles, even if they

sometimes wait for resource tiles? In the examples provided, the answer is yes, if the total

concentration of resource tiles is sufficiently low.

As an illustration, we compare the growth rate of the matched delay assemblies of

Fig. 4a with the growth rate of the ‘‘null assembly’’, shown in Fig. 4a, both growing in the

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(a) (b)

(c) (d)

Fig. 4 Time varying tile concentrations and adapted assemblies. Assemblies that grow from a set of tilesthat encode boolean functions. Shown here are four environments in which the concentrations of tileschange over time, and an assembly that is well adapted to each environment. Selection favors crystals thatcorrectly predict the concentration of sometimes scarce resource tiles. (a) Over time, measurement tiles andresources tiles have exactly the same time-varying concentrations. A well-adapted assembly asks for aresource tile with the same label as the measurement tile that was received. (b) Opposite resource andmeasurement tiles have the same concentrations. A well-adapted assembly uses a single computation tilethat inverts the input from the measurement tile, so that the opposite resource tile can bind. (c) The kind ofmeasurement tiles that are abundant is perfectly correlated with the type of resource tiles that will beabundant after a fixed time delay. A well-adapted assembly stores information about previous tiles andpasses it across the assembly. If the time necessary to pass the type of measurement tile across the assemblyis approximately the same as the delay in the correlation, the assembly will ask for the abundant resourcetiles. A wider assembly produces a longer delay. (d) Measurement tiles provide no information, but resourcetiles change regularly in a way that an assembly can predict. The assembly shown here uses a small counter(Cook et al. 2004) to change its resource tile request every 8 layers. The left-most row remembers thecurrent choice until the counter sends the message to change it. The rightmost column disregards theinformation provided by the measurement tiles


123

delay environment of Fig. 4c. Under what circumstances would a delay assembly of width

k grow faster than the null assembly? If tiles are added at an average of 1 per second, and

resource tiles overall are D times less common than computation and measurement tiles,

the growth rate of a perfectly synchronized delay assembly of width k ‘‘J’’ gates is

2k + D + 1 per zig-zag. (A width k assembly is suited to a delay of Dt = k(2k + D + 1)

seconds.) Assuming that the type of resource tile that is abundant switches on average

every s seconds, with Dt� s, the null assembly adds a zig-zag every 2(D + 1) seconds on

average2. When 2k < D + 1 (and Dt is such that the assembly provides the right delay), the

delay assembly grows faster. Clearly, this is true for large enough D.

Similarly, for the binary counter assembly (Fig. 4d), the growth rate of the perfectly

synchronized counter of width k is 2k + D + 1 seconds per zig-zag3. The growth rate of the

null assembly is D + 1 seconds per zig-zag half the time for an average growth rate of

2D + 2 seconds per zig-zag. Thus, when 2k + D + 1 < 2D + 2, the counter assembly is

more fit. These examples illustrate that when D is large, that is, when the time spent adding

computation and measurement tiles is negligible compared with the time spent waiting for

resource tiles, performing the right computations can make an assembly more fit.

In order for a crystal that is good at predicting the current availability of resource tiles to

be fit, it must be able to pass along its fitness to its descendent crystals. In this section, we

have shown how every row of a crystal could contain the information for a simple program.

Thus, when a crystal splits, each descendent contains the same program, which will also be

executed. However, while the program is preserved by the descendents, the state of the

program is not. The descendent crystals may start by executing the program from many

steps ago. In some such cases, such as where a crystal is running a program to keep track of

delays, the descendent crystal would then not grow well until it resynchronizes with the

environment.

It might seem easier for an assembly to simply accept both kinds of prediction tiles.

However, the chemistry of the tile set forbids this—1 and 0 labels do not match, and

therefore cannot bind to each other. The fitness landscape is such that increases in fitness

can only be achieved by good predictions. Thus, our examples address the question we set

out to consider: how a fixed chemistry (a tile set) induces a selection pressure in which

crystals must compute in order to be fit.

5 Evolution of universal sensing and response

In the last section, we showed how a set of tiles that allow crystals to execute and replicate

a program could, in the right environment, select for crystals performing a useful com-

putation. However, the tiles shown in Fig. 3a cannot simulate a Turing machine, and

therefore cannot perform universal computation (Sipser 1997). Thus there are computa-

tions the tiles cannot express; such computations might be needed for accurate prediction

in some environments. In this section, following (Rothemund and Winfree 2000), we

describe a tile set that can simulate a Turing machine and how we can alter this tile set to

2 This estimate assumes that since Dt � s, at any particular time the resource and measurement tileconcentrations are uncorrelated. So half the time, the resource and measurement tiles are compatible, and thecrystal grows at a rate of D + 1 seconds per zig-zag; and half the time the resource and measurement tiles arenot compatible, and the crystal essentially doesn’t grow.3 Note that counters whose natural period is slightly less than the period of the environment will easilyremain synchronized with the environment, even when there is variation in the arrival times of the tilesbeing added.


123

perform the sensing of measurement tiles and binding of resource tiles described in Sect. 4.

Because the Turing machine can compute any desired function, crystals grown using this

tile set are capable of universal sensing and response. That is, in response to arbitrarily

complex relationships between measurement and resource tiles, the fittest programs can

also become arbitrarily complex.

A Turing machine consists of a long work tape which has a sequence of symbols written

on it, and a head that can be in a finite number of states. The head examines the work tape,

one symbol at a time, executing a series of movements and updates to the symbols written

on the work tape based on what it observes locally. At each state of the computation, the

head is in a particular state and at a particular position on the work tape. The state and the

symbol written on the work tape determine a symbol the head will replace the current work

tape symbol with, the direction the head will move in (either one step to the left or right)

and the next state for the head to enter.

The set of tiles that allows zig-zag ribbon assembly to simulate the execution of a

particular Turing machine4 are shown in Fig. 5. Each row of an assembly of these tiles

represents the work tape at a particular time point in the computation. The design of the tile

set is such that as assembly proceeds up the ribbon, most of the tape is copied from row to

row (Fig. 6a), but some cells are altered as directed by the Turing machine. For com-

pactness and efficiency, all tape manipulations performed during a single unidirectional run

of the Turing machine head occur in a single layer of the crystal. Thus, there is one layer

per reversal of the Turing machine head.

At each step of assembly, exactly one location on the top row of the assembly has a

place for a tile to join by two edges, and thus is available for assembly. At each such

location, one of three things happens. If the position where the tile is to be added is not the

position of the head, the attachment of a new tile will copy the tape from the last row.

When assembly is proceeding to the right, the left edge will present a ‘‘right’’ label, and a

tile can attach to a location along the top row if it directs assembly to the right and matches

the tape value presented by the cell directly beneath it. Likewise, a tile can attach as

assembly is proceeding to the left if it matches the ‘‘left’’ label presented by the right edge

and matches the tape value presented by the tape (Fig. 6a). These steps copy the values on

the work tape.

At locations where the head is present, values on the work tape can change in sub-

sequent layers. Here, a matching tile type not only matches the old work tape value and the

direction of assembly, but also the current head state. The output edges of such a matching

tile direct the new head state and the new value of the work tape at this location (Fig. 6b). If

the tape value and head state direct the head to move in the direction of assembly, the

output edge in the direction of assembly (either to the left or right) encodes the new head

state. If the tape value and head state direct the head to move in the direction opposite to

the direction of assembly, the upward-facing edge of the attaching tile encodes the new

head state (Fig. 6c). Computation continues when assembly finishes in the current direction

and zig-zags back to the column where the tile attached (Figure 6d).

Because a zig-zag tile set exists for the simulation of an arbitrary Turing machine, such

a tile set exists for the simulation of a universal Turing machine. A universal Turing

machine is a Turing machine that can simulate the execution of any other Turing machine

4 While zig-zag assemblies copy layers on both growth fronts, for the purposes of our argument, we willassume that computation on zig-zag assemblies proceeds in only one direction (upward). While this is notnecessarily so for the tile sets we describe, growth can be restricted to one direction by using a transformedtile set (Winfree 2006) that uses extra tiles to prevent growth in the wrong direction.


123

when both the desired Turing machine and this Turing machine’s initial work tape are

encoded on the universal Turing machine’s work tape (Sipser 1997). Small universal

Turing machines exist that compute efficiently (Watanabe 1961) and simulate a Turing

machine by encoding it on one part of the tape and a work tape on another, as illustrated in

Fig. 7a.

It is not hard to imagine using the measurement and resource tiles described in Sect. 4

instead of the single kind of boundary tiles shown in Fig. 5 with the computation tiles that

can simulate a universal Turing machine. It would also be possible to add a few tiles to

input the measurement value and pass it onto the work tape, and likewise, to pass an output

from the Turing machine to the left edge of the assembly as the binding type to present in

order to bind a resource tile. The result would be assemblies with the structure shown in

Fig. 7b. With such a tile set, it could be possible for assemblies that computed and

responded to any desired correlation between measurement and resource tiles to grow.

Given an environment in which there is a complex correlation between the measurement

and resource tiles that can be computed by a Turing machine, under conditions where the

resource tiles are sufficiently rare, the fittest assembly would be one that exactly computed

this correlation and asked for the appropriate resource tiles.

If the correlation between measurement and resource tiles were very complex, it is

possible that by the time the assembly that exactly simulated the correlation finished

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Fig. 5 Tiles for the simulation of a Turing machine. A schematic for a set of tiles whose assembly cansimulate the computation of a particular Turing machine on a tape consisting of a row of tiles. A tile typeexists to copy each possible work tape value for the cases when assembly is proceeding both to the left andto the right. For each head state and input combination, tile types are needed for encoding the cases wherethe head is continuing to move in the same direction as the last step, is switching directions on the next step,or has switched directions on the last step. For head state and work tape combinations where the head shouldmove to the right after computation, the directions of assembly will be the reverse of those on the tilesshown here. Left and right double tiles form the boundary of the work tape. When more work tape is neededto the left, an L-shaped tile can provide one extra work tape location on the next row (converse tiles exist toextend the work tape to the right). For a Turing machine that has k work tape symbols and s head states, 3ks+ 2k + 2s + 2 tiles are needed to simulate it. (A slightly more complex tile set can also shrink the width ofthe zig-zag when less work tape is needed.)


123

computing it, the concentrations of the resource tiles would already have changed. In such

an environment, a complex program for prediction may not be selected for. However, one

can imagine an almost identical environment where that program would be selected for, in

which the time the measurement tiles were present, the wait between presenting the rel-

evant resource tiles, and the time the resource tiles were present were all longer by a

constant factor. In an environment where the pattern of increasing and decreasing con-

centrations of measurement and resource tiles was sufficiently slowed down, the complex

assembly would eventually be able to predict the right resource tiles in time, and would

therefore be fit.

Since Turing-computable correlations have arbitrarily long history dependence, it may

be difficult or impossible for a descendent crystal whose growth edge is at a previous state

in the computation to resynchronize after crystal reproduction by fragmentation. A simple

augmentation to the tile set rectifies this problem: a special head state triggers a ‘‘budding’’

process that duplicates and forks the computation (Fig. 8). This produces two growth fronts

that have the exact same computation state as well as program.

A generalization of this type of algorithmically controlled morphogenesis was previ-

ously described in detail for investigating the Kolmogorov complexity of self-assembled

. . . . . .

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Fig. 6 Computation with the Turing machine title set. (a) At a position where the head is not present, anattaching tile copies the value on the work tape. (b) When the head is an input from the right, computationproceeds because the attaching tile matches in the input state of the head. The output edge of this tilecontains the next head state. If computation continues to the left, this head state is outputted to the left. Thevalue on the work tape is updated according to the new value specified by the old work tape value and headstate. (c) If a head state and work tape value direct the head to move in the opposite direction as assembly isproceeding, the head state is output up, and computation occurs on the assembly of the next level, byattaching a tile that contains the head state on its bottom edge. Computation can then continue in theopposite direction (or switch directions again). (d) An example of the progress of the head (shaded) of aTuring machine on a zig-zag that is simulating a Turing machine. White cells simply copy work tape valuesfrom the row beneath them


123

shapes (Soloveichik and Winfree 2004). Since some shapes that could be created using this

tile set may have selective advantage (particularly shear-prone shapes, for example, may be

good at reproducing), evolution using such a tile set would result in selection for assem-

blies that contain programs for certain shapes. Descendents of these crystals will compute

the same shape (perhaps starting from a random step in the computation) so that

descendent crystals will also have selective advantage5.

6 Discussion

While it is possible to increase the complexity of selective pressures by increasing the

complexity of tile sets used for growth, we’ve suggested here that tile sets exist that allow

crystals to encode and run programs that can predict arbitrarily complex changes in the

environment. Evolution using such a tile set as a medium could produce very complex

sequences that grow well because they are particularly good at predicting changes in

growth conditions. This suggests that crystal growth chemistries logically can support

open-ended evolution with arbitrarily complex fitness landscapes.

Our argument that some assemblies will be more fit relies only on their growth rates,

which are determined by how effective the assemblies are at predicting the availability of

resource tiles. However, as Sect. 3 illustrates, to be fit, assemblies must not only grow

quickly, but also shear frequently. Not enough is known about shearing frequencies to

make confident predictions, as these rates are surely dependent on the kind of forces that

produce shearing in practice. However, it seems safe to assume that wider zig-zag

assemblies would shear less frequently than thin ones in most cases. If this were the case,

of two assemblies that predicted the availability of resource tiles equally well, the thinner

Programspecification

Work tape Programspecification

Work tapeinputLoad

Store

output

andpresent

(a) (b)

Fig. 7 Using a tile set capable of universal computation for metabolic control. (a) The construction shownin Fig. 5 shows how to translate an arbitrary Turing machine into a tile set that simulates it. Applying thisconstruction to a universal Turing machine (UTM) yields a tile set that forms zig-zag ribbons of the typeshown here: the program being executed by the UTM is stored in the ribbon separate from the work tapeused by the program. (b) A tile set that simulates a universal Turing machine can be combined withboundary tiles that are either resource or measurement tiles. Tiles can also be added to load the input frommeasurement tiles onto the work tape and to output the desired binding site for the resource tile. With such atile set, crystals that simulate a Turing machine that correctly predicts the concentrations of resource tilesover time would be most fit

5 In cases where a complete shape is necessary for selective advantage, it is also possible to use a con-struction where crystals can grow forward as well as backward, so that the parent crystal can grow back thepiece of the shape that was lost during splitting (Winfree 2006).


123

assembly would be more fit. Such an effect could actually have the effect of favoring

correct and concise programs for prediction of resource tile concentrations, i.e., it is a

natural implementation of Occam’s razor. This scenario is somewhat akin to Solomonoff

inference (Solomonoff 1964) and Levin search (Levin 1973) in that short programs that

produce correct results are found by biased random search.

However, our arguments in this paper are limited: they neglect several important real

life features of crystal growth. For example, while the model used in this paper prohibits

any tile that matches fewer than two bonds from attaching, such attachments occur at a rate

dependent on the physical conditions of assembly. If this error rate is too large, two things

can happen. First, particularly large programs that take a long time to run may have

difficulty accurately computing without mistakes, and there may be a preference for so-

called robust programs (if such programs exist for the tile set used), which compute an

answer correctly even with a couple of assembly errors. Second, it may be possible for

assemblies that ask for the absent resource tile to eventually attach the available but non-

matching resource tile. Also, we have also not considered the effects of backward growth

on the fitness landscape of crystals. For some tile sets, backward growth is not deter-

ministic and quickly leads to a configuration that cannot be continued except by making an

error, in which case backward growth stalls and can be neglected. For other tile sets,

backward growth is deterministic and produces the same class of patterns that can be

produced by forward growth. Yet another simplification we have made is to ignore sto-

chastic effects during assembly and how they affect the time at which an assembly asks for

a resource tile. Again, we do not expect such considerations do significantly change our

general conclusions.

Despite these and other limitations to our study as presented here, we believe that the

qualitative features of the mechanisms that we describe should be preserved in a more

realistic model of crystal growth. While the tile sets we described here are too complex to

implement experimentally at this time, it is not unreasonable to believe that much simpler

state before forkcopies of the computation

computationproceeds

before forkcomputationstate of

computationproceeds

computationoriginal

Fig. 8 Reproduction by budding. While splitting is simple, assemblies might also program theirreproduction time by ‘‘budding’’ instead of breaking. Budding assemblies may reproduce by firstassembling a structure that allows computation to proceed in two directions.


123

tile sets could produce complex adaptations in response to resource limitations. Our initial

investigations into whether the ideas here could be tested in experiments suggest that tile

sets containing as few as ten tile types could in fact exhibit complex evolution in an

environment with constant tile concentrations, as will be described elsewhere. Such a small

tile set and environment would be amenable to laboratory experiments to test whether non-

trivial crystal genotypes that efficiently use available resources could evolve. While

selection for larger and larger genotypes require more and more accurate copying of

information (Eigen 1971) and selection based on smaller fitness differences, error-resistant

tile sets (Winfree and Bekbolatov 2004; Chen and Goel 2004; Reif et al. 2004), which

contain more tiles but copy more accurately could be used. In principle these tile sets can

reduce error rates as much as is desired.

One might think that for the fittest species in a given environment to be arbitrarily

complex, the environment must be equally complex. But it is not immediately clear

whether this is actually the case. Since there is no fastest algorithm for some functions

(Blum 1967), this would seem to imply that in a crystal growth environment that requires

crystals to compute such a function, evolutionary pressure favoring speed would lead to

ever-increasing complexity. On other hand, faster programs are in general larger. So while

it is possible that evolutionary pressure might favor faster algorithms, the crystal must also

be wider in order to contain the complex program and copying the program slows down the

crystal’s growth. This trade-off is also present in biological, in vitro, and artificial evo-

lution.

It is also interesting to ask about the minimal requirements for open-ended evolution of

crystals. Is it possible that there is a ‘‘universal’’ environment where tile concentrations are

constant or vary in a simple periodic pattern, but open-ended evolution is still possible? To

answer such a question, it may be worth considering effects that are beyond the simple

model of crystal growth and replication kinetics used in this paper. We considered only

fitness of species for growth in the exponential phase, where resources are not depleted by

other crystals, and where there is no interaction between crystals. Relaxing these

assumptions may allow other interesting routes to complexity. For example, could com-

petition between crystals for tiles lead to an ‘‘arms-race’’ of more and more complex

strategies for growth? Conversely, might symbiosis between crystal programs lead to

interesting phenotypes not explored here? Might a crystal growth chemistry support par-

asitic crystals that evolve to grow by using existing crystals rather than single tiles?

Supporting this last possibility, initial experiments with DNA crystals indicate that joining

of crystals does occur (Ekani-Nkodo et al. 2004).

Acknowledgments We thank Andrew Turberfield, Paul Rothemund, Robert Barish, Ho-Lin Chen, andAshish Goel for insightful conversations and suggestions. This work was supported by NASA Grant No.NNG06GA50G.

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